The present invention adopts well known mathematical games of prior art, and provides their use in gifts, souvenirs, and ornaments, broadening mathematical game principle of operation in relation to graphics, number relationships, etc. FIG. 1 presents three examples of the prior art. FIG. 1a illustrates a paper-and-pencil game consisted of a triangular graph with four nodes on a side, which hereafter are designated by a side pattern. Initially the nodes are empty and the player is asked to fill in the numbers 1 to 9 in such a way that an equal sum is obtained upon addition of all the numbers on each side pattern. FIG. 1a shows a solution with 17 as the equal sum number. Other equal sum arrangements are possible as well, with either 17 or other number as the equal sum number.
FIG. 1b presents a magic pentagram, or star of fifth order, having five valley nodes and five vertex nodes, wherein a number appears at every node. The number arrangement provides a magic pentagram so that five straight-line-patterns have a relationship that an identical sum is obtained upon addition of all the numbers which appear on the nodes of said straight-line-pattern.
A two thousand years old prior art is shown in FIG. 1c, a 3×3 magic square, whereas a sum of 15 is obtained upon addition of a number triplet which appears in a row pattern or in a column pattern. In general, the challenge of N×N magic square is to get the equal sum arrangement using consecutive numbers, classically 1 to N2, without repetitions. Famous mathematician Euler and his followers have shown that for most of the natural numbers N, several of the magic square solutions may be obtained using the concepts of Latin and Graeco-Latin symbol arrangement. Those concepts are clearly defined in U.S. Pat. No. 3,189,350 (issued Jun. 15, 1965) to Hopkins:                “In the Latin square . . . having N squares on a side, a series of N symbols, such as Latin letters, are so arranged that no symbol occurs twice in any row or in any column. Many such arrangements are possible, and from this evolved a more complicated square in which two different arrangements of Latin squares are superimposed so that two symbols appear in each small square. It is a consequence of this arrangement that in addition to the two different solutions of the Latin square, a further solution is provided in that no two-symbol combination appears twice in any row or in any column. As a convenience the two Latin squares are made up from two different families of symbols, such as Greek letters and Latin letters, from which the term Graeco-Latin arises . . . ”.        
In other words, Graeco-Latin arrangement of N×N square simultaneously provides for three relationships:                1. The indices of a first type of indices are Latin arranged.        2. The indices of a second type of indices are Latin arranged.        3. No combination of an index of the first type of indices and an index of the second type of indices appears twice in the whole N×N square.        
FIG. 1d presents a Graeco-Latin of the A, B, C Latin letters and the α,β, γ Greek letters. Once that arrangement is achieved, the numbers in the magic square of FIG. 1c are obtained upon execution of three steps:                1. Replacement of the Latin letters A, B, C with 1, 2,3, respectively.        2. Replacement of the Greek letters α,β, γ by 0,3,6, respectively.        3. Addition of every number corresponding to a Latin letter in a small square to the respective number corresponding to the Greek letter in the same small square.        
Weekend newspaper editions suggest a Sudoko challenge, in which one has to complete absent 1-9 numbers in a 9×9 square, in order to obtain both a 9×9 Latin square arrangement over row and column patterns, and a Latin arrangement over nine 3×3 square patterns. Sometimes, several Sudoko challenges are being offered in a varying level, covering a full spectrum of newspaper reader ability.
U.S. Pat. No. 4,128,243 (issued Dec. 5, 1978) to Pulejo describes a magic square puzzle composed of five pieces which should be arranged in a 4×4 Latin square of the four domino-like indices of one to four dots.
U.S. Pat. No. 6,206,372 (issued Mar. 27, 2001) to Harris deals with a 5×5 magic square puzzle game and suggests element coloring to facilitate a desired magic square solution.